| Sparse | $. _ _ _ | Sparse |
|
$.y converts a dense array to sparse, and
conversely $.^:_1 y converts a sparse array to
dense. The identities f -: f&.$. and f -: f&.($.^:_1) hold for any function f , with the possible exception of those (like overtake {.) which use the sparse element as the fill. |
0$.y applies $. or $.^:_1 as
appropriate; that is, converts a dense array to sparse and a
sparse array to dense. 1$.sh;a;e produces a sparse array. sh specifies the shape. a specifies the sparse axes; negative indexing may be used. e specifies the "zero" element, and its type determines the type of the array. The argument may also be sh;a (e is assumed to be a floating point 0) or just sh (a is assumed to be i.#sh -- all axes are sparse -- and e a floating point 0). 2$.y gives the sparse axes (the a part); (2;a)$.y (re-)specifies the sparse axes; (2 1;a)$.y gives the number of bytes required for (2;a)$.y; (2 2;a)$.y gives the number of items in the i part or the x part for the specified sparse axes a (that is, #4$.(2;a)$.y). 3$.y gives the sparse element (the e part); (3;e)$.y respecifies the sparse element. 4$.y gives the index matrix (the i part). 5$.y gives the value array (the x part). 6$.y gives the flag (the flag part). 7$.y gives the number of non-sparse entries in array y; that is, #4$.y or #5$.y. 8$.y removes any completely "zero" value cells and the corresponding rows in the index matrix. The inverse of n&$. is (-n)&$. . |
] d=: (?. 3 4$2) * ?. 3 4$100
0 75 0 53
0 0 67 67
93 0 51 83
] s=: $. d convert d to sparse and assign to s
0 1 | 75
0 3 | 53 the display of s gives the indices of the
1 2 | 67 "non-zero" cells and the corresponding values
1 3 | 67
2 0 | 93
2 2 | 51
2 3 | 83
d -: s d and s match
1
o. s p times s
0 1 | 235.619
0 3 | 166.504
1 2 | 210.487
1 3 | 210.487
2 0 | 292.168
2 2 | 160.221
2 3 | 260.752
o. d p times d
0 235.619 0 166.504
0 0 210.487 210.487
292.168 0 160.221 260.752
(o. s) -: o. d function results independent of representation
1
0.5 + o. s
0 1 | 236.119
0 3 | 167.004
1 2 | 210.987
1 3 | 210.987
2 0 | 292.668
2 2 | 160.721
2 3 | 261.252
<. 0.5 + o. s
0 1 | 236
0 3 | 167
1 2 | 210
1 3 | 210
2 0 | 292
2 2 | 160
2 3 | 261
(<. 0.5 + o. s) -: <. 0.5 + o. d
1
d + s function arguments can be dense or sparse
0 1 | 150
0 3 | 106
1 2 | 134
1 3 | 134
2 0 | 186
2 2 | 102
2 3 | 166
(d + s) -: 2*s familiar algebraic properties are preserved
1
(d + s) -: 2*d
1
+/ s
0 | 93
1 | 75
2 | 118
3 | 203
+/"1 s
0 | 128
1 | 134
2 | 227
|. s reverse
0 0 | 93
0 2 | 51
0 3 | 83
1 2 | 67
1 3 | 67
2 1 | 75
2 3 | 53
|."1 s
0 0 | 53
0 2 | 75
1 0 | 67
1 1 | 67
2 0 | 83
2 1 | 51
2 3 | 93
|: s transpose
0 2 | 93
1 0 | 75
2 1 | 67
2 2 | 51
3 0 | 53
3 1 | 67
3 2 | 83
$ |: s
4 3
$.^:_1 |: s $.^:_1 converts a sparse array to dense
0 0 93
75 0 0
0 67 51
53 67 83
(|:s) -: |:d
1
, s ravel; a sparse vector
1 | 75
3 | 53
6 | 67
7 | 67
8 | 93
10 | 51
11 | 83
$ , s
12
Representation| sh | Shape, $y . Elements of the shape must be less than 2^31 , but the product over the shape may be larger than 2^31 . |
| a | Axe(s), a vector of the sorted sparse (indexed) axes. |
| e | Sparse element ("zero"). e is also used as the fill in any overtake of the array. |
| i | Indices, an integer matrix of indices for the sparse axes. |
| x | Values, a (dense) array of usually non-zero cells for the non-sparse axes corresponding to the index matrix i . |
| flag | Various bit flags. |
] d=: (?. 3 4$2) * ?. 3 4$100
0 75 0 53
0 0 67 67
93 0 51 83
] s=: $. d
0 1 | 75
0 3 | 53
1 2 | 67
1 3 | 67
2 0 | 93
2 2 | 51
2 3 | 83
The shape is 3 4 ;
the sparse axes are 0 1 ;
the sparse element is 0;
the indices are the first two columns of numbers
in the display of s ; and the values are
the last column.| imax =: _1+2^31 | the largest internal integer |
| rank =: #@$ | rank |
| type =: 3!:0 | internal type |
| 1 = rank sh | vector |
| sh -: <. sh | integral |
| imax >: #sh | at most imax elements |
| (0<:sh) *. (sh<:imax) | bounded by 0 and imax |
| 1 = rank a | vector |
| a e. i.#sh | bounded by 0 and rank-1 |
| a -: ~. a | elements are unique |
| a -: /:~ a | sorted |
| 0 = rank e | atomic |
| (type e) = type x | has the same internal type as x |
| 2 = rank i | matrix |
| 4 = type i | integral |
| (#i) = #x | as many rows as the number of items in x |
| ({:$i) = #a | as many columns as there are sparse axes |
| (#i) <: */a{sh | # rows bounded by product over sparse axes lengths |
| imax >: */$i | # elements is bounded by imax |
| (0<:i) *. (i <"1 a{sh) | i bounded by 0 and the lengths of the sparse axes |
| i -: ~.i | rows are unique |
| i -: /:~ i | rows are sorted |
| (rank x) = 1+(#sh)-#a | rank equals 1 plus the number of dense axes |
| imax >: */$x | # elements is bounded by imax |
| (}.$x)-:((i.#sh)-.a){s | item shape is the dimensions of the dense axes |
| (type x) e. 1 2 4 8 16 32 | internal type is boolean, character, integer, real, complex, or boxed |
] d=: (0=?. 2 3 4$3) * ?. 2 3 4$100
13 0 0 0
21 4 0 0
0 0 0 0
3 5 0 0
0 0 6 0
0 0 0 0
] s=: $. d convert d to sparse and assign to s
0 0 0 | 13
0 1 0 | 21
0 1 1 | 4
1 0 0 | 3
1 0 1 | 5
1 1 2 | 6
d -: s match is independent of representation
1
2 $. s sparse axes
0 1 2
3 $. s sparse element
0
4 $. s index matrix; columns correspond to the sparse axes
0 1 0
0 1 1
1 0 0
1 0 1
1 1 2
5 $. s corresponding values
13 21 4 3 5 6
] u=: (2;2)$.s make 2 be the sparse axis
0 | 13 21 0
| 3 0 0
|
1 | 0 4 0
| 5 0 0
|
2 | 0 0 0
| 0 6 0
4 $. u index matrix
0
1
2
5 $. u corresponding values
13 21 0
3 0 0
0 4 0
5 0 0
0 0 0
0 6 0
] t=: (2;0 1)$.s make 0 1 be the sparse axes
0 0 | 13 0 0 0
0 1 | 21 4 0 0
1 0 | 3 5 0 0
1 1 | 0 0 6 0
7 {. t take
0 0 | 13 0 0 0
0 1 | 21 4 0 0
1 0 | 3 5 0 0
1 1 | 0 0 6 0
$ 7 {. t
7 3 4
7{."1 t take with rank
0 0 | 13 0 0 0 0 0 0
0 1 | 21 4 0 0 0 0 0
1 0 | 3 5 0 0 0 0 0
1 1 | 0 0 6 0 0 0 0
0 = t
0 0 | 0 1 1 1
0 1 | 0 0 1 1
1 0 | 0 0 1 1
1 1 | 1 1 0 1
3 $. 0 = t the sparse element of 0=t is 1
1
+/ , 0 = t
18
+/ , 0 = d answers are independent of representation
18
0 { t from
0 | 13 0 0 0
1 | 21 4 0 0
_2 (<1 2 3)}t amend
0 0 | 13 0 0 0
0 1 | 21 4 0 0
1 0 | 3 5 0 0
1 1 | 0 0 6 0
1 2 | 0 0 0 _2
s=: 1 $. 20 50 1000 75 366
$ s 20 countries, 50 regions, 1000 salespersons,
20 50 1000 75 366 75 products, 366 days in a year
*/ $ s the product over the shape can be greater than 2^31
2.745e10
r=: ?. 1e5 $ 1e6 revenues
i=: ?. 1e5 5 $ $ s corresponding locations
s=: r (<"1 i)} s assign revenues to corresponding locations
7 {. ": s the first 7 rows in the display of s
0 0 5 30 267 | 128133 the first row says that for country 0, region 0,
0 0 26 20 162 | 319804 salesperson 5, product 30, day 267,
0 0 31 37 211 | 349445 the revenue was 128133
0 0 37 10 351 | 765935
0 0 56 6 67 | 457449
0 0 66 54 120 | 38186
0 0 71 74 246 | 515473
+/ , s total revenue
|limit error the expression failed on ,s because it would
| +/ ,s have required a vector of length 2.745e10
+/@, s total revenue
5.00289e10 f/@, is supported by special code
+/+/+/+/+/ s total revenue
5.00289e10
+/^:5 s
5.00289e10
+/^:_ s
5.00289e10
+/ r
5.00289e10
+/"1^:4 s total revenue by country
0 | 2.48411e9
1 | 2.55592e9
2 | 2.55103e9
3 | 2.52089e9
4 | 2.49225e9
5 | 2.45682e9
6 | 2.52786e9
7 | 2.45425e9
8 | 2.48729e9
9 | 2.50094e9
10 | 2.51109e9
11 | 2.59601e9
12 | 2.49003e9
13 | 2.58199e9
14 | 2.44772e9
15 | 2.47863e9
16 | 2.46455e9
17 | 2.5568e9
18 | 2.43492e9
19 | 2.43582e9
t=: +/^:2 +/"1^:2 s total revenue by salesperson
$t
1000
7{.t
0 | 4.58962e7
1 | 4.81548e7
2 | 3.97248e7
3 | 4.89981e7
4 | 4.85948e7
5 | 4.69227e7
6 | 4.22094e7
Sparse Linear Algebraf=: }. @ }: @ (,/) @ (,."_1 +/&_1 0 1) @ i. f 5 indices for a 5 by 5 tri-diagonal matrix 0 0 0 1 1 0 1 1 1 2 2 1 2 2 2 3 3 2 3 3 3 4 4 3 4 4 s=: (?. 13$100) (<"1 f 5)} 1 $. 5 5;0 1 $s 5 5The phrase 1$.5 5;0 1 makes a sparse array with shape 5 5 and sparse axes 0 1 (sparse in both dimensions); <"1 f 5 makes boxed indices; and x (<"1 f 5)}y amends by x the locations in y indicated by the indices (scattered amendment).
s 0 0 | 13 0 1 | 75 1 0 | 45 1 1 | 53 1 2 | 21 2 1 | 4 2 2 | 67 2 3 | 67 3 2 | 93 3 3 | 38 3 4 | 51 4 3 | 83 4 4 | 3 ] d=: $.^:_1 s the dense representation of s 13 75 0 0 0 45 53 21 0 0 0 4 67 67 0 0 0 93 38 51 0 0 0 83 3 ] y=: ?. 5$80 10 60 36 42 17 y %. s 1.27885 _0.0883347 0.339681 0.202906 0.0529263 y %. d answers are independent of representation 1.27885 _0.0883347 0.339681 0.202906 0.0529263 s=: (?. (_2+3*1e5)$1000) (<"1 f 1e5)} 1 $. 1e5 1e5;0 1 $ s s is a 1e5 by 1e5 matrix 100000 100000 y=: ?. 1e5$1000 ts=: 6!:2 , 7!:2@] time and space for execution ts 'y %. s' 0.28 5.2439e6 0.28 seconds; 5.2 megabytes (Pentium 266 Mhz)Implementation Status
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Notes: